Proposition 12

("associativity" of the tensor $\varepsilon$-product of graded algebras of the same types). *With the notation of Proposition 10 of no. 7, suppose that all the $\Delta_i$ are equal to the same monoid $\Delta_0$; let $(J_\lambda)_{\lambda \in L}$ be a partition of I. With the notation of Proposition 11 of no. 7, suppose that the right hand side of formula (32) (no. 7) depends only on the sums $\alpha''_\lambda = \sum_{i \in J_\lambda} \alpha_i, \beta''_\mu = \sum_{j \in J_\mu} \beta_j$, for every ordered pair $(\lambda, \mu)$ of distinct indices, all $\alpha'_\lambda \in \Delta'_\lambda$ and all $\beta'_\mu \in \Delta'_\mu$; let $\varepsilon''_{\lambda \mu}(\alpha''_\lambda, \beta''_\mu)$ denote the right hand side of (32). *Then $(\varepsilon''_{\lambda \mu})$ is a system of commutation factors over the family $(\Delta''_\lambda)_{\lambda \in L}$, where $\Delta''_\lambda = \Delta_0$ for all $\lambda \in L$. *If $E''_\lambda$ is the graded tensor $\varepsilon$-product of type $\Delta_0$ of the family $(E_i)_{i \in J_\lambda}$, there exists one and only one isomorphism of graded algebras of type $\Delta_0$, $w : \varepsilon'' \bigotimes_{\lambda \in L} E''_\lambda \to \varepsilon \bigotimes_{i \in I} E_i$, such that

provided that total orderings are chosen on the $J_\lambda$ and on I as described in no. 7, Proposition 11.

By the hypothesis, for $\gamma, \delta$ in $\Delta_0$, $\varepsilon''_{\lambda \mu}(\gamma, \delta) = \varepsilon_{i_0 j_0}(\gamma, \delta)$ for some $i_0 \in J_\lambda$ and some $j_0 \in J_\mu$, as is seen by considering the elements $\alpha'_\lambda = (\alpha_i)_{i \in J_\lambda}$ and $\beta'_\mu = (\beta_j)_{j \in J_\mu}$ such that $\alpha_{i_0} = \gamma, \alpha_i = 0$ for $i \neq i_0$, $\beta_{j_0} = \delta, \beta_j = 0$ for $j \neq j_0$; it follows immediately that the $\varepsilon''_{\lambda \mu}$ form a system of commutation factors. The rest of the proof is then analogous to that of Proposition 11 (no. 7) and is left to the reader.

Note that the additional hypotheses of Proposition 12 are fulfilled when $\Delta_0 = \mathbf{Z}$ and that $(\varepsilon_{ij})$ is, either the trivial system of factors $(\varepsilon_{ij}(\alpha_i, \beta_j)) = 1$ for all $i, j$, or the system of factors defined by $\varepsilon_{ij}(\alpha_i, \beta_j) = (-1)^{\alpha_i \beta_j}$; in the latter case, the right hand side of formula (32) is equal to $(-1)^\gamma$, where